I have the same problem. The "Integrate Variables" filter in Paraview is apparently rather disfunctional: it only works for lines and surfaces. Its failure on volumes can be easily demonstrated by integrating over a 1x1x1 cube. However, the underyling VTK library does support volume integration, so it could be just a matter of invoking the VTK function through the Paraview Python interface.

Any Paraview nerds that can help out with this? I already tried the Paraview-users mailing list, but got no response.